M5L21f.txt

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So let us assume that we have an atom which
starts in the ground state, but now we excite the atom.
And we excite it by a short pulse.
And this can be a pulse, which is has
a pulse angle between 0 and pi.
And depending what the pulse angle is,
we have the ground state, and we'll
mix something of the excited state.
And in case of a pi pulse, we have 100% in the excited state.
So our atomic wave function is this.
The excited state-- let's just assume at the photons
are on resonance.
We know how to deal with off-resonant lasers.
So, therefore, the phase evolution
is e to the i omega0 t.
But now, and this is what coherence is about,
there is a very specific phase.
And this phase, phi, comes from the laser.
If you excite the atom with a laser beam,
but it has a phase shift, then the atomic wave function
is phase shifted.
Because every amplitude you had mixed into the ground state
in form of an excited state, was driven by the operator--
the dipole operator e-- and e, the electric field,
has a phase of the laser beam.
Sure, there may be-- and kind of all other trivial factors
I've set to 1 here-- but there is
the phase of the laser which directly
is imprinted into the phase of the wave function.

So this is what the atom does, what the laser
beam does to the atom.
So we have now an atom, which is partially excited,
and it carries an imprint off the phase of the laser.
And now, after the laser pulse is over,
the photonic part of the function is a vacuum.
We have no photon in our cavity.
And now we wait and we allow spontaneous emission.
And spontaneous emission is nothing else
than a time evolution with the operator I just
discussed with you.
So after spontaneous emission, one as we know for sure,
the atom is in the ground state.
Let me write down the result. If you apply the operator which
couples to the electromagnetic field,
and we assume only co-rotating terms here of-- let's just
neglect counterrotating rotating terms, which
can become-- which [INAUDIBLE] resonance are irrelevant.
What happens is, we are now propagating this wave function.
And the ground state-- with the ground state of the photon,
does nothing.
However, the excited state with 0 photon-- we discussed it.
Excited state with 0 photon will actually
do Rabi oscillation-- ground state with 1 photon;
excited state with 0 photons.
And so we have now our knowledge from the vacuum Rabi
oscillation . that this is part of the wave function does
nothing.
This part of the wave function undergoes single photon vacuum
Rabi oscillations.
And if you start out with the superposition of ground 0
and excited 0, well this is a superposition principle
of quantum mechanics, we can just propagate this part
and we can propagate that part.
So what I suggest is when we-- really
at the fundamental level, and I'll
discuss spontaneous emission, we allow the part, excited atom
empty cavity, to undergo half a vacuum Rabi oscillation.
And then the excited state is in the ground state.
And the photon state has 1 photon.
It's a completely coherent Rabi oscillation.
It just allows half a cycle to evolve.
And the result of that is-- well,
nothing happened to this part.
And the Rabi oscillations have now
taken us to the 1 photon state.

And it has just swapped excited 0 to ground state 1 photon.

So let me write down something which is really remarkable.
And then we discuss on Monday-- next Monday,
we discuss how we would really measure the phase.
But just look at the two expressions I've underlined.
And this tells us that the quantum state
of the atom as a two-level system-- ground excited
state with all the phase factors--
has been now exactly matched on the quantum
state of the cavity.
So if you regard ground and excited state
as a two-level system, every quantum mechanical,
subtly of the atomic system has now disappeared.
It's in the ground state.
But everything which was coherent,
which was a phase, which was interesting
about the atomic system has been transferred
to the photon field.
Yes?
[INAUDIBLE], do you want to have an alpha e there also?
Yes, please.
Thank you.
When I said everything, I meant everything.

So this is what I meant, that this
is the most fundamental aspect of spontaneous emission.
The quantum state of the atom has been perfectly
matched, perfectly mapped, onto the photon field.

And the one thing we have to discuss on Wednesday
is the role of the phase phi.
We started out by phi being the phase of the laser.
And if the laser is in a coherent state--
and I will talk about on Wednesday,
in a [INAUDIBLE] measurement, we can measure phi
to any accuracy you want.
We can determine the phase of the laser
in [INAUDIBLE] experiment.
This phase phi has been now perfectly imprinted
into a two-level system for the atom.
And now it appears mapped into a two-level system
for the photons-- the two-level system between 0 photons and 1
photons.
But if we are now doing a measurement,
either on the atomic system or on the photonic system,
we are limited in the accuracy at which we can retrieve phi.
And this is what you're going to discuss on Wednesday.
And this is what I referred too as the fundamental limit
of spontaneous emission.
Because we have not lost any coherence here,
it's just if the phase is only imprinted in one particle.
One particle quantum physics sets us a limitation
how well we can read out the phase phi.

Any questions?